Solution of a Riccati equation for the design of an observer contracting a Riemannian distance

We propose a method to design an intrinsic observer guaranteeing that the Riemannian distance between the estimate it generates and the state of the system is decreasing in time, at least locally. The design relies on the existence of a Riemannian metric, the Lie derivative of which along the system vector field is negative in the space tangent to the level sets of the output function. We show that, at least when the system is uniformly strongly infinitesimally observable (i.e., each time-varying linear system resulting from the linearization along a solution to the system satisfies a uniform observability property), there exists such a metric and it can be obtained as a solution to an algebraic-like Riccati equation. For such systems, we propose also an algorithm to numerically approximate the metric by griding the space and integrating ordinary differential equations.